While I didn't exactly enjoy my course taught that way, as a human plugging in those methods manually, it was sort of interesting from the perspective of later being a fairly heavy user of computer algebra systems (CASs). The bag-of-tricks approach they teach in school is really how such systems work in practice, and tons of practical problems will be solved that way, either with you doing it by hand, or using software that does it for you. Software like Maxima, Maple, Mathematica, Sage, etc. consists of a huge pile of case analysis techniques and methods that pattern-match on specific equation forms that can be solved with the method in question. A CAS does feel a bit more satisfying because it feels like the pile of techniques is at least being given some kind of formalization and order, versus me just trying to remember them. Although the amount of order is not quite as much as one might like; even using a CAS there's still a lot poring over documentation to find the function that works in your case, which will go a bit faster if you remember enough of the textbook methods to recognize what you're looking for.

That's not to say this is anything like what mathematicians do, especially PhD mathematics researchers. But a lot of applied mathematics in engineering is not that far off from what's taught in a university ODE class.